Vibrations and Waves

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Vibrations and Waves

• Chapter 12

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12.1 Simple Harmonic Motion
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Remember

• Elastic Potential Energy (PEe) is the energy
  stored in a stretched or compressed elastic
  object
• Gravitational Potential Energy (PEg) is the
  energy associated with an object due to its
  position relative to Earth

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Useful Definitions

• Periodic Motion A repeated motion. If it is
  back and forth over the same path, it is called
  simple harmonic motion.
• Examples Wrecking ball, pendulum of clock
• Simple Harmonic Motion Vibration about an
  equilibrium position in which a restoring force
  is proportional to the displacement from
  equilibrium
• http//www.ngsir.netfirms.com/englishhtm/SpringSHM
  .htm

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Useful Definitions

• A spring constant (k) is a measure of how
  resistant a spring is to being compressed or
  stretched.
• (k) is always a positive number
• The displacement (x) is the distance from
  equilibrium.
• (x) can be positive or negative. In a spring-mass
  system, positive force means a negative
  displacement, and negative force means a positive
  displacement.

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Hookes Law

• Hookes Law for small displacements from
  equilibrium
• Felastic -(kx)
• Spring force -(spring constant x displacement)
• This means a stretched or compressed spring has
  elastic potential energy.
• Example Bow and Arrow

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Example Problem

• If a mass of 0.55 kg attached to a vertical
  spring stretches the spring 2.0 cm from its
  original equilibrium position, what is the spring
  constant?

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Example Answer

• Given m 0.55 kg x -0.020m
• g 9.81 k ?
• Fg mg 0.55 kg x 9.81 5.40 N
• Hookes Law F -kx
• 5.40 N -k(-0.020m) k 270 N/m

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12.2 Measuring simple harmonic motion
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Useful Definitions

• Amplitude the maximum angular displacement from
  equilibrium.
• Period the time it takes to execute a complete
  cycle of motion
• Symbol T SI Unit second (s)
• Frequency the number of cycles or vibrations
  per unit of time
• Symbol f SI Unit hertz (Hz)

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Formulas - Pendulums

• T 1/f or f 1/T
• The period of a pendulum depends on the string
  length and free-fall acceleration (g)
• T 2pv(L/g)
• Period 2p x square root of (length divided by
  free-fall acceleration)

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Formulas Mass-spring systems

• Period of a mass-spring system depends on mass
  and spring constant
• A heavier mass has a greater period, thus as mass
  increases, the period of vibration increases.
• T 2pv(m/k)
• Period 2p x the square root of (mass divided by
  spring constant)

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Example Problem- Pendulum

• You need to know the height of a tower, but
  darkness obscures the ceiling. You note that a
  pendulum extending from the ceiling almost
  touches the floor and that its period is 12s.
  How tall is the tower?

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Example Answer

• Given T 12 s g 9.81 L ?
• T 2pv(L/g)
• 12 2 pv(L/9.81)
• 144 4p2L/9.81
• 1412.64 4p2L
• 35.8 m L

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Example Problem- Mass-Spring

• The body of a 1275 kg car is supported in a frame
  by four springs. Two people riding in the car
  have a combined mass of 153 kg. When driven over
  a pothole in the road, the frame vibrates with a
  period of 0.840 s. For the first few seconds, the
  vibration approximates simple harmonic motion.
  Find the spring constant of a single spring.

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Example answer

• Total mass of car people 1428 kg
• Mass on 1 tire 1428 kg/4 357 kg
• T 0.840 s
• T 2pv(m/k)
• K(4p2m)/T2
• K (4p2(357 kg))/(0.840 s)2
• k 2.00104 N/m

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12.3 Properties of Waves
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Useful Definitions

• Crest the highest point above the equilibrium
  position
• Trough the lowest point below the equilibrium
  position
• Wavelength ? the distance between two adjacent
  similar points of the wave

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Wave Motion

• A wave is the motion of a disturbance.
• Medium the material through which a disturbance
  travels
• Mechanical waves a wave that requires a medium
  to travel through
• Electromagnetic waves do not require a medium to
  travel through

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Wave Types

• Pulse wave a single, non-periodic
• disturbance
• Periodic wave a wave whose source is some form
  of periodic motion
• When the periodic motion is simple harmonic
  motion, then the wave is a SINE WAVE (a type of
  periodic wave)
• Transverse wave a wave whose particles vibrate
  perpendicularly to the direction of wave motion
• Longitudinal wave a wave whose particles vibrate
  parallel to the direction of wave motion

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• Transverse Wave
• Longitudinal Wave

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Speed of a Wave

• Speed of a wave frequency x wavelength
• v f?
• Example Problem
• The piano string tuned to middle C vibrates with
  a frequency of 264 Hz. Assuming the speed of
  sound in air is 343 m/s, find the wavelength of
  the sound waves produced by the string.
• v f?
• 343 m/s (264 Hz)(?)
• 1.30 m ?

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• The Nature of Waves Video 220

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12.4 Wave Interactions
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Constructive vs Destructive Interference

• Constructive Interference individual
  displacements on the same side of the equilibrium
  position are added together to form the resultant
  wave
• Destructive Interference individual
  displacements on the opposite sides of the
  equilibrium position are added together to form
  the resultant wave

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• Wave Interference Demo

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When Waves Reach a Boundary

• At a free boundary, waves are reflected

• At a fixed boundary, waves are reflected and
  inverted

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Standing Waves

• Standing wave a wave pattern that results when
  two waves of the same frequency, wavelength, and
  amplitude travel in opposite directions and
  interfere
• Node a point in a standing wave that always
  undergoes complete destructive interference and
  therefore is stationary
• Antinode a point in a standing wave, halfway
  between two nodes, at which the largest amplitude
  occurs

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• Ruben's Tube Video 157